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Chapter 12: Problem 56

Water to run a Pelton wheel is supplied by a penstcck of length \(\ell\) and diameter \(D\) with a friction factor \(f\). If the only losses associated with the flow in the penstock are due to pipe friction. show that the maximum power output of the turbine occurs when the nozzle diameter, \(D_{1},\) is given by \(D_{1}=D /(2 f \ell / D)^{1 / 4}\)

Short Answer

Expert verified
After solving the problem, it's demonstrated that the diameter for the maximum power output for a Pelton wheel turbine when the losses are due to pipe friction is given by \(D_{1}=D /(2 f \ell / D)^{1 / 4}\).

Step by step solution

01

Understand and list known quantities

Given: diameter of penstock \(D\), length of penstock \(\ell\), friction factor \(f\) and nozzle diameter \(D_1\). We are required to show \(D_{1}=D /(2 f \ell / D)^{1 / 4}\). The formula for the loss of energy due to friction is given as \(h_f = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g}\) where \(h_f\) is the head loss, \(v\) is the velocity, \(L\) is the length of the pipe, and \(g\) is the acceleration due to gravity.
02

Derive power output

The total hydraulic energy available at the nozzle is \(\rho g\ h\) where \(h = H - h_f\). The mechanical power output from a turbine is given by \(P = \rho g Q h\) where \(Q\) is the flow rate.
03

Express Q in terms of \(D_1\)

The flow rate \(Q\) can be expressed in terms of nozzle diameter \(D_1\) and velocity at the nozzle \(v_1\) as \(Q = \frac{π D_1^2 v_1}{4}\). The velocity at the nozzle exit \(v_1\) is given by \(v_1=\sqrt{2 g h}\). Substitute for \(v_1\) and \(h\) in the expression for power output. You should now have the power output equation as a function of nozzle diameter \(D_1\). It involves \(D\), \(f\), \(D_1\) and \(\ell\).
04

Maximize power output

To determine the maximum power output, take the derivative of the power equation with respect to \(D_1\) and set it to zero. Solve this equation for \(D_1\) to find the nozzle diameter that maximizes power output.
05

Verify the derived expression

The solution from the previous step should match the given expression \(D_{1}=D /(2 f \ell / D)^{1 / 4}\). This proves that the maximum power output of the turbine occurs when the nozzle diameter is as given by the derived equation.

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Most popular questions from this chapter

A prototype fan has a \(20-f\) diameter, an inlet pressure of 14.40 psia, an inlet temperature of \(70^{\circ} \mathrm{F},\) and a speed of 90 rpm. \(\mathrm{A}\) \(\frac{1}{10}-\operatorname{scal} \mathrm{e}\) model of the fan has the same inlet pressure and emperature. an inlet power of \(1.24 \mathrm{hp}\), a flow rate of \(220 \mathrm{ft}^{3} / \mathrm{min}\), and a speed of \(1800 \mathrm{rpm} .\) Find the corresponding input power and flow rate of the prototype fan. Neglect Reynolds number effects.

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A centrifugal pump having an impeller diameter of \(1 \mathrm{m}\) is to be constructed so that it will supply a head rise of 200 mat a flowrate of \(4.1 \mathrm{m}^{3} / \mathrm{s}\) of water when operating at a speed of \(1200 \mathrm{rpm} .\) To study the characteristizs of this pump, a \(1 / 5\) scale, geometrizally similar model operated at the same speed is to be tested in the laboratory. Determine the required model discharge and head rise. Assume that both model and prototype operate with the same effciency (and therefore the same flow coefficient)

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